Find the mass and center of mass of the lamina that occupies the region D and has the given density function rho. D is the triangular region with vertices (0, 0), (2, 1), (0, 3); rho(x, y) = 2(x + y)

mass (m) = 4 kg

              center of mass coordinate: (15.75,4.5)

Step-by-step explanation: As a surface, a lamina has 2 dimensions (x,y) and a density function.

The region D is shown in the attachment.

From the image of the triangle, lamina is limited at x-axis: 0≤x≤2

At y-axis, it is limited by the lines formed between (0,0) and (2,1) and (2,1) and (0.3):

Points (0,0) and (2,1):

y =

y =

Points (2,1) and (0,3):

y =

y = -x + 3

Now, find total mass, which is given by the formula:

Calculating for the limits above:

where a = -x+3

m = 2(-4+6)

m = 4

Mass of the lamina that occupies region D is 4.

Center of mass is the point of gravity of an object if it is in an uniform gravitational field. For the lamina, or any other 2 dimensional object, center of mass is calculated by:

and are moments of the lamina about x-axis and y-axis, respectively.

Calculating moments:

For moment about x-axis:

Now to find the x-coordinate:

x =

x =

x = 15.75

For moment about the y-axis:

63

To find y-coordinate:

y =

y =

y = 4.5

Center mass coordinates for the lamina are (15.75,4.5)

the solution is 12x-6y